> Had we decided to adopt a different rule, arithmetic would still have worked, and we'd never have needed imaginary numbers. As other sibling comments have sa

> Had we decided to adopt a different rule, arithmetic would still have worked, and we'd never have needed imaginary numbers. As other sibling comments have said, what would your arithmetic look like that still worked, but with a different 'rule' (which is a rule just in the sense of a consequence of the axioms, not in the sense of an arbitrary choice)? If you wanted, for example, (3 + -2)(3 + -2) to equal (3)(3) + (3)(-2) + (-2)(3) + (-2)(-2), then you'd have a hard time getting that equality to hold.